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Theorem dfdm3 4796
Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfdm3  |-  dom  A  =  { x  |  E. y <. x ,  y
>.  e.  A }
Distinct variable group:    x, y, A

Proof of Theorem dfdm3
StepHypRef Expression
1 df-dm 4619 . 2  |-  dom  A  =  { x  |  E. y  x A y }
2 df-br 3988 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32exbii 1598 . . 3  |-  ( E. y  x A y  <->  E. y <. x ,  y
>.  e.  A )
43abbii 2286 . 2  |-  { x  |  E. y  x A y }  =  {
x  |  E. y <. x ,  y >.  e.  A }
51, 4eqtri 2191 1  |-  dom  A  =  { x  |  E. y <. x ,  y
>.  e.  A }
Colors of variables: wff set class
Syntax hints:    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156   <.cop 3584   class class class wbr 3987   dom cdm 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-br 3988  df-dm 4619
This theorem is referenced by:  csbdmg  4803
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